Question 536702
If the graphs of all the equations in the system turn out to be the same line, then every point on one graph will be a point of the graph for the other equations, and since each point represents a solution, all the solutions on one graph are also solutions on the other graph.
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If there are an infinite number of solutions, you will find that each of the equations in the system can be shown to be identical to the other equations.
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For example: suppose the system consists of the following equations:
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{{{y = 10x + 4}}}
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{{{-20x + 2y = 8}}}
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{{{-30x - 12 = -3y}}}
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All of these equations have an infinite number of solutions. They are all the same equation, just in different forms.
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The first equation is in Slope-Intercept form {{{y = mx + b}}} in which m (the multiplier of the x) is the slope and b (the constant) is the point on the Y-axis where the graph crosses. 
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Let's leave the first equation {{{y = 10x + 4}}} in the Slope-Intercept form. Now, lets rearrange the second equation and get it into the same form.
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Start with the second equation:
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{{{-20x + 2y = 8}}}
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Move the -20x to the right side by adding 20x to both sides to get:
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{{{2y = 20x + 8}}}
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Now solve for y by dividing both sides (all terms) by 2. The result is:
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{{{y = 10x + 4}}}
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This shows that although the second equation looked different from the first equation, it can be put into the Slope-Intercept form and reduced to exactly the same equation as the first one. Therefore, having the same slope and the same intercept on the Y-axis, it has the same graph as the first and therefore every point on its graph represents the same solution on the first graph. 
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Finally, let's look at the third equation:
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{{{-30x - 12 = -3y}}}
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Let's manipulate it into Slope-Intercept form. We can solve for y by dividing all terms on both sides by -3 and in doing so we get:
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{{{10x + 4 = y}}}
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Now just transpose the sides and you have:
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{{{y = 10x + 4}}}
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Again this is identical to the first (and also the second) equation. So it has infinitely many solutions in common with both those equations. It also has a graph with the same slope and same y-intercept as the other two equations.
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As long as you rearrange and manipulate the equations of a system into some sort of common form, you should be able to reduce them to the same equation if they have an infinite number of solutions in common. I just find it easy to use the Slope-Intercept form. Plus you can do the following analyses:
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If you get two linear equations into the Slope-Intercept form of {{{y = mx + b}}} and you see that they have different slopes (that is the coefficients or multipliers of x are different in the two equations) then the graphs will cross at only one point. So there will be one unique, common solution only.
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If you find that the slopes are the same in both equations, but the constants are different between the two equations, the graphs are separate but parallel. Therefore, there will be no common solution.
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And as we showed above, if the slopes are the same and the constants are the same, then the graphs are identical and there are an infinite number of common solutions.
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Hope this discussion helps you to understand the situation a little better.
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